p-adic motivic cohomology in

arithmetic geometry

Wies lawa Nizio l

Motivic cohomology

• over fields:

DMk - the derived category of mixed motives

over a field k; constructions by Levine, Vo-

evodsky (1990’s)

HiM(X, Z(n)) = ExtiMMk(Z(0), h(X)⊗ Z(n))

= HomDMk(Z(0), ZX(n)[i])

' Hi(X, Z(n))

Hi(X, Z(n)) - Bloch higher Chow groups

Rationally:

HiM(X, Z(n))⊗Q ' grnγ K2n−i(X)⊗Q

- γ-graded pieces of K-theory

• over Dedekind domains:

fundations are still missing; use Bloch higher

Chow groups

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Motivic vs arithmetic cohomologies

• Zariski vs etale motivic cohomology

Motivic cohomology can be defined both in the

Zariski and the etale topology. The change of

topology map:

ρi,n : Hi(X, Z/m(n))→ Hi(Xet,Z/m(n))

Beilinson, Lichtenbaum, Quillen, Thomason:

ρi,n is an isomorphism for large n

• Etale motivic vs arithmetic cohomologies

Realizations:

etale∼←− Hi(Xet,Z/m(n))

∼−→ syntomicyolog de Rham-Witt

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Milnor K-theory

k - a field, char(k) = p; the Milnor K groups:

KM∗ (k) = T ∗Z(k×)/(x⊗ (1− x)|x ∈ k − 0,1)Milnor KM∗ /m = Galois cohomology:

• (m, p) = 1; Kummer theory:

KM1 (k)/m ' k∗/k∗m ∼→ H1(ket, µm).

Cup product gives the Galois symbol map

KMn (k)/m→ Hn(ket, µ

⊗nm ).

Conjecture (Bloch-Kato) The Galois sym-

bol map is an isomorphism.

Voevodsky (2001, 2002): true for m = 2n;

conditionally for general m

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Milnor KM∗ /m = Galois cohomology:

• m = pr > 0; Bloch-Gabber-Kato:

dlog : KMn (k)/pr ∼→ H0(ket, ν

nr ).

The (etale) logarithmic de Rham-Witt sheaf:

νnr = 〈x ∈WrΩ

nX|x = dlog x1 ∧ . . . ∧ dlog xn〉

0→ νn· →W·ΩnX

F−1→ W·ΩnX → 0

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Motivic cohomology

X - a separated scheme over a field;

Bloch higher Chow groups:

Hi(X, Z(n)) =Hi(Z(n)(X))

Hi(X,Z/m(n)) =Hi(Z(n)(X)⊗ Z/m)

Z(n)(X) := zn(X,2n− ∗) is the complex:

• algebraic r-simplex:

4r ' ArZ = SpecZ[t0, . . . , tr]/(

∑ti − 1)

• zn(X, i) ⊂ zn(X×4i) - the free abelian group

generated by irreducible codimension n subva-

rieties of X ×4i meeting all faces properly.

• the chain complex

zn(X, ∗) : zn(X,0)← zn(X,1)← zn(X,2)←boundaries: restrictions of cycles to faces

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• Hi(X, Z(n)) = 0 for i > min2n, n + dimX

• Beilinson-Soule conjecture (still open):

Hi(X, Z(n)) = 0, i < 0

• H2n(X, Z(n)) ' CHn(X), Chow group

• for a field k, Hn(k, Z(n)) ' KMn (k)

• localization (difficult): for Zc

→ X ← U

→Hi−2c(Z,Z(n− c))→ Hi(X, Z(n))→Hi(U,Z(n))→ Hi+1−2c(Z, Z(n− c))→

• higher cycle classes (difficult):

ci,n : Hi(X, Z(n))→ Hi(X, n)

needed: weak purity and homotopy property

Question How to define cycle classes into

syntomic cohomology ?

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Etale motivic cohomology

X 7→ Z(n)(X) := zn(X,2n − ∗): sheaf in theetale topology; X separated, noetherian :

Hi(X, Z(n)) ' Hi(XZar,Z(n))

• X/k - smooth, k-perfect, char(k) = p > 0,Geisser-Levine (2000):

Hi+n(XZar,Z/pr(n)) ' Hi(XZar, νnr )

Hi+n(Xet,Z/pr(n)) ' Hi(Xet, νnr )

• X smooth, m invertible on X; cycle class

ceti,n : Hi(Xet,Z/m(n))∼→ Hi(Xet, µ

⊗nm )

Change of topology map:

ρi,n : Hi(XZar,Z/m(n))→ Hi(Xet,Z/m(n))

Example

ρ2n,n = clet : CHn(X)/m→ H2n(Xet, µ⊗nm )

– neither injective nor surjective

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Conjecture (Beilinson-Lichtenbaum)

ρi,n : Hi(X, Z/m(n))∼→ Hi(Xet,Z/m(n)), i ≤ n

• Suslin (2000): true for k = k, n ≥ dimX

• Suslin-Voevodsky, Geisser-Levine (2000):

Bloch-Kato ⇔ Beilinson-Lichtenbaum

- use Gersten resolution to pass from fields to

schemes:

0→ Hp(Z(n))→ ⊕

x∈X(0)

(ix)∗Hp(kx,Z(n))→⊕

x∈X(1)

(ix)∗Hp−1(kx,Z(n− 1))→

X(s) - points in X of codimension s

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• V - complete dvr, mixed characteristic (0, p),

perfect residue field; X/V - smooth scheme

Geisser (2004): Bloch-Kato mod p ⇒

(1) Beilinson-Lichtenbaum mod p is true:

ρi,n : Hi(X, Z/pr(n))∼→ Hi(Xet,Z/pr(n)), i ≤ n

(2) for X proper and i ≤ r < p− 1:

csyni,r : Hi(Xet,Z/pn(r))∼→ Hi(X, Sn(r))

– uses p-adic Hodge theory !

Sn(r) - the syntomic complex:

→ Hi(Xn, Sn(r))→ FrHidR(Xn)

1−φ/pr

−→ HidR(Xn)

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Algebraic K-theory

X- noetherian scheme; MX (resp. PX) – the

category of coherent (resp. locally free sheaves)

on X. MX → K′X, PX → KX: certain associ-

ated simplicial spaces.

The algebraic K and K ′ groups of X:

Ki(X) = πi(KX), Ki(X, Z/m) = πi(KX,Z/m),

K ′i(X) = πi(K′X), K ′i(X, Z/m) = πi(K′X,Z/m)

• K0(X) and K ′0(X) are the Grothendieck

groups of vector bundles and coherent sheaves

• k - field, KM1 (k) = K1(k) = k∗; product:

KMn (k)→ Kn(k)

that is an isomorphism for n ≤ 2

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• γ filtration (exterior powers of vector bun-

dles ): F ∗γ K∗(X)

grjγ K2j−i(X)⊗Q ' Hi(X, Q(j))

In fact, modulo small torsion

• Poincare duality: if X is regular then

Ki(X)∼→ K ′i(X)

• localization (easy): for Z → X ← U ,

→ K ′i+1(U)→ K ′i(Z)→ K ′i(X)→ K ′i(U)→

• higher Chern classes (easy):

ci,n : grnγ K2n−i(X)→ Hi(X, n)

Needed: the cohomology of BGL is the

right one (which holds for most p-adic co-

homologies)

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Etale K-theory

X 7→ K(X), X 7→ K/m(X) : presheaves of sim-

plicial spaces. Assume X is regular.

K∗(X, Z/m) ' H−∗(XZar,K/m).

If m is invertible on X then the etale K-theory

of Dwyer-Friedlander

K etj (X, Z/m) ' H−j(Xet,K/m).

Conjecture (Quillen-Lichtenbaum)

ρj : Kj(X, Z/m)∼→ K et

j (X, Z/m), j ≥ cdm Xet

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Atiyah-Hirzebruch spectral sequence

motivic cohomology ⇒ K-theory

X/k - smooth; Bloch-Lichtenbaum, Friedlander-

Suslin, Levine; Grayson-Suslin (1995-2003):

Es,t2 = Hs−t(X, Z/m(−t))⇒ K−s−t(X, Z/m)

• m invertible on X; Levine (1999):

Es,t2 = Hs−t(Xet,Z/m(−t))⇒ K et−s−t(X, Z/m)

Degenerates at E2 modulo small torsion ⇒ceti,j : grjγ K et

2j−i(X, Z/m)∼→ Hi(Xet, µ

⊗jm )

Beilinson-Lichtenbaum ⇒ Quillen-Lichtenbaum

• k - perfect, char(k) = p > 0, Geisser-Levine:

Kn(X, Z/pr) = 0 for n > dimX

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Application: p-adic Hodge theory (Niziol)

K - complete dvf, char = (0, p), V - ring ofintegers, k - perfect residue field; K - an alge-braic closure of K, GK - its Galois groupX/V - proper variety, XK - smooth

p-adic Hodge theory:

etale: Hi(XK,Qp) + GK-action

p-adic period map: m

de Rham : HidR(XK/K) + F ∗, φ, N

Methods:

• syntomic; Fontaine-Messing, Kato, Tsuji (1987-1998): period map = the cospecialization mapon the syntomic-etale site• almost-etale; Faltings (1989-2002)• motivic; Niziol (1998-2006): period map =the localization map in motivic cohomology

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The good reduction case

V = W - Witt vectors of k, X/V - smooth,

proper, relative dimension d

Crystalline conjecture: For i ≤ r < p − 1,

there exists a GK-equivariant period isomor-

phism

Hi(XK, µ⊗rn )

∼→ Fr(HidR(Xn/Vn)⊗B+

cr,n)pr=φ

' Hi(XV , Sn(r))

• crystalline ring of periods:

B+cr,n = H∗cr(Spec(V n)/Wn); F ∗, φ, GK

• de Rham cohomology:

HidR(Xn/Vn) ' H∗cr(X0/Vn), F ∗, φ

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Bloch-Kato ⇒ Crystalline conjecture

Assume that we have syntomic cycle maps

Hi(XV ,Z/pn(r))j∗−→∼ Hi(XK,Z/pn(r))

yρi,r

yρKi,r

Hi(XV ,et,Z/pn(r))j∗−→ Hi(XK,et,Z/pn(r))

ycsyni,r oyceti,r

Hi(XV , Sn(r))αcr

i,r←−− Hi(XK, µ⊗rpn )

• the key point: j∗ is an isomorphism: by

localization, the kernel and cokernel are con-trolled by Hi(Xk,Z/pn(r)), which is killed by

totally ramified extensions of V of degree pn

• ρKi,r is an isomorphism for r ≥ i: by Beilinson-

Lichtenbaum

• define the map αcri,r to make this diagram

commute.

• Notice: all the maps in the above diagram

are isomorphisms.

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Suslin ⇒ Crystalline conjecture

grrγ K2r−i(XV ,Z/pn)∼−→j∗

grrγ K2r−i(XK,Z/pn)yρ2r−i

yρK2r−i

grrγ K et2r−i(XV ,Z/pn) −→

j∗grrγ K et

2r−i(XK,Z/pn)ycsyni,r

yceti,r

Hi(XV , Sn(r))αcr

i,r←−− Hi(XK, µ⊗rpn ).

• the key point: j∗ is an isomorphism (as

before)

•ρK2r−i is an isomorphism modulo small torsion

for 2r − i ≥ 2d: by Suslin

•ceti,r is an isomorphism modulo small torsion:

by the Atiyah-Hirzebruch spectral sequence

• define αcri,r to make this diagram commute

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The semistable reduction case

X×/V × - proper, semistable reduction, no mul-tiplicities (more generally: log-smooth=toroidal)

Semistable conjecture There exists

αst : H∗(XK,Qp)⊗QpBst ' H∗cr(X×0 /W0)⊗W Bst

preserving GK-action, N , F ∗, φ

• the period ring Bst: GK, φ, N , F ∗• H∗cr(X×0 /W0)[1/p] (analogue of limit Hodgestructures); φ, N , F ∗:

K ⊗W H∗cr(X×0 /W0) ' H∗dR(XK/K)

Corollary

H∗(XK,Qp) ' F0(H∗cr(X×0 /W0)⊗Bst)N=0,φ=1

Have: for i ≤ r

Fr(Hicr(X

×0 /W0)⊗Bst)

N=0,φ=pr

' Hi(X×V

, SQp(r))

Hi(X×V

, SQp(r)) - log-syntomic cohomology

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Semistable period maps: for r large enougha compatible family

αni,r : Hi(XK, µ⊗r

pn )→ Hi(X×V

, Sn(r))

Main difficulty: the model XV is in generalsingular

• higher Chow groups behave badly

• the localization map j∗ in K-theory is dif-ficult to understand

Solution:

• each X×V ′, finite extension V ′/V , can be

desingularized by a log-blow-up Y × → X×V ′,

Y - regular

• log-blow-up does not change the log-syntomiccohomology, so to define the maps αn

i,r wecan work with the regular models Y ×

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Suslin ⇒ Semistable conjecture

grrγ K2r−i(Y,Z/pn)∼−→j∗

grrγ K2r−i(YK,Z/pn)ycsyni,r ρ2r−i o

yceti,rρ2r−i

Hi(Y ×, Sn(r))αn

i,r←−− Hi(YK, µ⊗rpn ).

• the key point: j∗ is an isomorphism for

2r − i > d + 1: use the localization sequence

→ K ′j(Yk,Z/pn)→ K ′j(Y,Z/pn)j∗→ K ′j(YK,Z/pn)

and Geisser-Levine: K ′j(Yk,Z/pn) = 0 for j > d

Corollary (Niziol, 2006): there exists a unique

period map

αsti,r : Hi(XK,Qp(r))→ Hi(X×

V, SQp

(r))

compatible with the etale and syntomic higher

Chern classes. In particular, the syntomic, al-

most etale and motivic period maps are equal.

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Ideal picture

Not yet in existence !

Hi(Y ×,Z/pn(r))j∗−→∼ Hi(YK,Z/pn(r))

yρi,r

yρKi,r

Hi(Y ×et ,Z/pn(r))j∗−→ Hi(YK,et,Z/pn(r))

ycsyni,r oyceti,r

Hi(Y ×, Sn(r))αn

i,r←−− Hi(Y, µ⊗rpn )

Question Can we define limit motivic coho-mology Hi(Y ×,Z/pn(r)) such that• the localization map j∗ is an isomorphism• the change of topology map ρi,r is an iso-morphism for r ≥ i

• the log-syntomic cycle classes csyni,r are iso-morphisms for p− 1 > r ≥ i

Recent work: Levine (2005), Ayoub, Niziol(2006)

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